狭义相对论和电磁学在定义为从时空到仿射空间的映射的参考系中的重新表述
Reformulation of Special Relativity and Electromagnetism in terms of Reference Frames defined as maps from spacetime onto affine spaces
摘要：
集合$M$上的参考系由3维欧几里德空间$E$、从$M$到$E$的函数、1维仿射空间$A$和从$M$到$A$的函数给出。该定义允许牛顿力学、狭义相对论和电磁学以框架而不是坐标的形式直观地和无坐标地表述。特别是，一套图表的假设$M$（地图集）被取代的一套框架的假设。然后，通过选择仿射坐标，可以重新获得图表。此外，单位作为正空间的元素进入理论，我们得到了一个几何的和明显的单位独立的重新表述。每一个框架都允许我们用一个4维乘积空间来识别时空，并且我们可以推广可微流形的定义，使得这些框架形成一个图集。牛顿力学和狭义相对论之间的唯一区别是假设的过渡函数类型--仿射空间之间的伽利略变换或庞加莱变换。这使我们能够在论文的第二章中突出共同点和差异。例如，我们用数学术语定义了速度互易性，并证明了这两种变换的现象。本章最后讨论了惯性系和加速系。在第三章中，我们把注意力限制在狭义相对论和电磁学的重新表述上，并着重强调协变性。世界线作为时空的特殊子集被引入，世界线的真时被定义为世界线上的仿射结构，电磁学的重新表述基于微分形式w.r.t.一个框架。
英文摘要：
A reference frame on a set $M$ is given by a 3-dimensional euclidean space $E$, a function from $M$ to $E$, a 1-dimensional affine space $A$ and a function from $M$ to $A$. The definition allows an intuitive and coordinate-free formulation of Newtonian mechanics, special relativity and electromagnetism in terms of frames instead of coordinates. In particular, the postulate of a set of charts on $M$ (an atlas) is replaced by the postulate of a set of frames. Then the charts can be re-obtained through the choice of affine coordinates. In addition, units enter the theory as elements of positive spaces and we obtain a geometric and manifestly unit-independent reformulation. Each frame allows us to identify spacetime with a 4-dimensional product-space and we can generalize the definition of differentiable manifolds such that the frames turn out to form an atlas. Then the only difference between Newtonian mechanics and special relativity is the assumed type of transition functions - Galilean transformations or Poincaré transformations between affine spaces. This allows us to highlight the common features and differences in the second chapter of the paper. For example, we define velocity reciprocity in mathematical terms and prove the phenomenon for both kind of transformations. The chapter concludes with a discussion of inertial and accelerated frames. In the third chapter we restrict our attention to the reformulation of special relativity and electromagnetism with a strong emphasis on covariance. World lines are introduced as particular subsets of spacetime, the proper time of world line is defined as an affine structure on the world line and the reformulation of electromagnetism is based on the representation of differential forms w.r.t. a frame.
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